A319056 -id:A319056 - cp's OEIS frontend

A323304 Heinz numbers of integer partitions that cannot be arranged into a matrix with equal row-sums and equal column-sums.

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105

Offset: 1

Gus Wiseman, Jan 13 2019

nonn

The first term of this sequence absent from A106543 is 144.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Index entries for sequences related to Heinz numbers

Complement of A323306.
Cf. A006052, A007016, A008480, A056239, A112798, A120733, A319056, A321719, A321721.
Cf. A323300, A323302, A323305, A323347, A323348,  A323349.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Select[Range[2,1000],Select[ptnmats[#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]

A323348 Number of integer partitions of n whose parts cannot be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 13, 17, 27, 36, 54, 66, 99, 128, 169, 221, 295, 367, 488, 610, 779, 993, 1253, 1525, 1955, 2426, 2986, 3684, 4563, 5519, 6840, 8298, 10097, 12298, 14874, 17716, 21635, 26002, 31105, 37081, 44581, 52916, 63259, 74852, 88703, 105543, 124752, 145740, 173522, 203999, 239737, 280424, 329929

Offset: 0

Gus Wiseman, Jan 13 2019

nonn

			The a(8) = 17 integer partitions:
  (53), (62), (71),
  (332), (422), (431), (521), (611),
  (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111).

Chai Wah Wu, Table of n, a(n) for n = 0..59

A000041(n) = A323347(n) + a(n).
Cf. A006052, A007016, A120733, A319056, A319066, A321719.
Cf. A323300, A323302, A323304, A323306, A323349.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Length[Select[IntegerPartitions[n],Select[ptnmats[Times@@Prime/@#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]],{n,10}]

a(17)-a(53) from Chai Wah Wu, Jan 15 2019

A321699 MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 19, 27, 32, 49, 53, 64, 81, 113, 128, 131, 151, 161, 165, 169, 225, 243, 256, 311, 343, 361, 512, 719, 729, 1024, 1291, 1321, 1619, 1937, 1957, 2021, 2048, 2093, 2117, 2187, 2197, 2257, 2401, 2805, 2809, 3375, 3671, 4096, 6561

Offset: 1

Gus Wiseman, Dec 27 2018

nonn

A multiset multisystem is a finite multiset of finite multisets. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is uniform if all parts have the same size, and regular if all vertices appear the same number of times. For example, {{1,1},{2,3},{2,3}} is uniform, regular, and spans an initial interval of positive integers, so its MM-number 15463 belongs to the sequence.

			The sequence of all uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:
    1: {}
    2: {{}}
    3: {{1}}
    4: {{},{}}
    7: {{1,1}}
    8: {{},{},{}}
    9: {{1},{1}}
   13: {{1,2}}
   15: {{1},{2}}
   16: {{},{},{},{}}
   19: {{1,1,1}}
   27: {{1},{1},{1}}
   32: {{},{},{},{},{}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   64: {{},{},{},{},{},{}}
   81: {{1},{1},{1},{1}}
  113: {{1,2,3}}
  128: {{},{},{},{},{},{},{}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  165: {{1},{2},{3}}
  169: {{1,2},{1,2}}
  225: {{1},{1},{2},{2}}
  243: {{1},{1},{1},{1},{1}}
  256: {{},{},{},{},{},{},{},{}}
  311: {{1,1,1,1,1,1}}
  343: {{1,1},{1,1},{1,1}}
  361: {{1,1,1},{1,1,1}}
  512: {{},{},{},{},{},{},{},{},{}}
  719: {{1,1,1,1,1,1,1}}
  729: {{1},{1},{1},{1},{1},{1}}

Cf. A005176, A007016, A112798, A302242, A306021, A319056, A319189, A320324, A321698, A321717, A322554, A322703, A322833.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[normQ[primeMS/@primeMS[#]],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A321725 Irregular triangle read by rows where T(n,k) is the number of d X d non-normal semi-magic squares with d = A027750(n,k) and sum of all entries equal to n.

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 3, 24, 1, 120, 1, 4, 21, 720, 1, 5040, 1, 5, 282, 40320, 1, 55, 362880, 1, 6, 6210, 3628800, 1, 39916800, 1, 7, 120, 2008, 202410, 479001600, 1, 6227020800, 1, 8, 9135630, 87178291200, 1, 231, 153040, 1307674368000, 1, 9, 10147

Offset: 1

Gus Wiseman, Nov 18 2018

A non-normal semi-magic square is a nonnegative integer square matrix with all row sums and column sums equal to d, for some d|n.

			Triangle begins:
   1
   1   2
   1   6
   1   3  24
   1 120
   1   4  21 720
The T(6,2) = 4 semi-magic squares (zeros not shown):
  [3  ] [2 1] [1 2] [  3]
  [  3] [1 2] [2 1] [3  ]
The T(6,3) = 21 semi-magic squares (zeros not shown):
  [2    ] [2    ] [2    ] [1 1  ] [1 1  ] [1 1  ] [1 1  ]
  [  2  ] [  1 1] [    2] [1 1  ] [1   1] [  1 1] [    2]
  [    2] [  1 1] [  2  ] [    2] [  1 1] [1   1] [1 1  ]
.
  [1   1] [1   1] [1   1] [1   1] [  2  ] [  2  ] [  2  ]
  [1 1  ] [1   1] [  2  ] [  1 1] [2    ] [1   1] [    2]
  [  1 1] [  2  ] [1   1] [1 1  ] [    2] [1   1] [2    ]
.
  [  1 1] [  1 1] [  1 1] [  1 1] [    2] [    2] [    2]
  [2    ] [1 1  ] [1   1] [  1 1] [2    ] [1 1  ] [  2  ]
  [  1 1] [1   1] [1 1  ] [2    ] [  2  ] [1 1  ] [2    ]

Chai Wah Wu, Table of n, a(n) for n = 1..60
Wikipedia, Magic square
Index entries for sequences related to magic squares

Cf. A006052, A007016, A027750, A120732, A319056, A319616.

Cf. A321718, A321719, A321721, A321722, A321724.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[k]==Union[Last/@#],SameQ@@Total/@prs2mat[#],SameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5},{k,Divisors[n]}]

T(n, A000005(n)) = n!. Sum_k T(n,k) = A321719(n). - Chai Wah Wu, Jan 15 2019

a(15)-a(48) from Chai Wah Wu, Jan 15 2019

Edited by Peter Munn, Mar 05 2025

A322531 Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 13, 15, 16, 17, 29, 31, 32, 33, 41, 43, 47, 51, 55, 59, 64, 67, 73, 79, 83, 85, 93, 101, 109, 113, 123, 127, 128, 137, 139, 149, 155, 157, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 233, 241, 249, 255, 256, 257, 269, 271

Offset: 1

Gus Wiseman, Dec 14 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
All entries are themselves squarefree numbers (except the powers of 2).
The first odd term not in this sequence but in A302521 is 141, which is the MM-number (see A302242) of {{1},{2,3}}.

			The sequence of all integer partitions whose parts all have the same number of prime factors and whose product of parts is a squarefree number begins: (), (1), (2), (1,1), (3), (1,1,1), (5), (6), (3,2), (1,1,1,1), (7), (10), (11), (1,1,1,1,1), (5,2), (13), (14), (15), (7,2), (5,3), (17), (1,1,1,1,1,1).

Cf. A003963, A005117, A038041, A056239, A073576, A112798, A302242, A302505, A306017, A319056, A319169, A320324, A321717, A321718, A322526, A322528.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[SameQ@@PrimeOmega/@primeMS[#],SquareFreeQ[Times@@primeMS[#]]]&]

A322703 Squarefree MM-numbers of strict uniform regular multiset systems spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 7, 13, 15, 19, 53, 113, 131, 151, 161, 165, 311, 719, 1291, 1321, 1619, 1937, 1957, 2021, 2093, 2117, 2257, 2805, 3671, 6997, 8161, 10627, 13969, 13987, 14023, 15617, 17719, 17863, 20443, 22207, 22339, 38873, 79349, 84017, 86955, 180503, 202133

Offset: 1

Gus Wiseman, Dec 27 2018

nonn

A multiset multisystem is a finite multiset of finite multisets. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is uniform if all parts have the same size, regular if all vertices appear the same number of times, and strict if there are no repeated parts. For example, {{1,1},{2,3},{2,3}} is uniform and regular but not strict, so its MM-number 15463 does not belong to the sequence. Note that the parts of parts such as {1,1} do not have to be distinct, only the multiset of parts.

			The sequence of all strict uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:
      1: {}
      2: {{}}
      3: {{1}}
      7: {{1,1}}
     13: {{1,2}}
     15: {{1},{2}}
     19: {{1,1,1}}
     53: {{1,1,1,1}}
    113: {{1,2,3}}
    131: {{1,1,1,1,1}}
    151: {{1,1,2,2}}
    161: {{1,1},{2,2}}
    165: {{1},{2},{3}}
    311: {{1,1,1,1,1,1}}
    719: {{1,1,1,1,1,1,1}}
   1291: {{1,2,3,4}}
   1321: {{1,1,1,2,2,2}}
   1619: {{1,1,1,1,1,1,1,1}}
   1937: {{1,2},{3,4}}
   1957: {{1,1,1},{2,2,2}}
   2021: {{1,4},{2,3}}
   2093: {{1,1},{1,2},{2,2}}
   2117: {{1,3},{2,4}}
   2257: {{1,1,2},{1,2,2}}
   2805: {{1},{2},{3},{4}}
   3671: {{1,1,1,1,1,1,1,1,1}}
   6997: {{1,1,2,2,3,3}}
   8161: {{1,1,1,1,1,1,1,1,1,1}}
  10627: {{1,1,1,1,2,2,2,2}}
  13969: {{1,2,2},{1,3,3}}
  13987: {{1,1,3},{2,2,3}}
  14023: {{1,1,2},{2,3,3}}
  15617: {{1,1},{2,2},{3,3}}
  17719: {{1,2},{1,3},{2,3}}
  17863: {{1,1,1,1,1,1,1,1,1,1,1}}

Cf. A005117, A007016, A112798, A302242, A306017, A306021, A319056, A319189, A319190, A320324, A321698, A321699, A322554, A322833.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322789 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 4, 3, 2, 2, 4, 7, 6, 4, 2, 2, 4, 10, 8, 4, 3, 7, 3, 4, 12, 8, 4, 2, 2, 6, 32, 35, 31, 18, 6, 2, 2, 4, 21, 10, 4, 4, 47, 29, 4, 5, 49, 72, 19, 5, 2, 2, 6, 81, 170, 71, 24, 6, 2, 2, 6, 138, 478, 296, 32, 6, 4, 429, 76, 4, 4, 64, 14, 4

Offset: 1

Gus Wiseman, Dec 26 2018

A multiset partition is uniform if all parts have the same size.

			Triangle begins:
  1
  2  2
  2  2
  3  4  3
  2  2
  4  7  6  4
  2  2
  4 10  8  4
  3  7  3
  4 12  8  4
Non-isomorphic representatives of the multiset partitions counted under row 6:
{123456}           {112233}           {111222}           {111111}
{123}{456}         {112}{233}         {111}{222}         {111}{111}
{12}{34}{56}       {123}{123}         {112}{122}         {11}{11}{11}
{1}{2}{3}{4}{5}{6} {11}{22}{33}       {11}{12}{22}       {1}{1}{1}{1}{1}{1}
                   {11}{23}{23}       {12}{12}{12}
                   {12}{13}{23}       {1}{1}{1}{2}{2}{2}
                   {1}{1}{2}{2}{3}{3}

Andrew Howroyd, Table of n, a(n) for n = 1..338 (rows 1..75)

Row sums are A319056. First column is A000005.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306017, A306018, A319190, A319612, A322784, A322785, A322787, A322792.

Terms a(28) and beyond from Andrew Howroyd, Feb 03 2022

Name edited by Peter Munn, Mar 05 2025

A322833 Squarefree MM-numbers of strict uniform regular multiset multisystems. Squarefree numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 41, 43, 47, 51, 53, 55, 59, 67, 73, 79, 83, 85, 93, 97, 101, 103, 109, 113, 123, 127, 131, 137, 139, 149, 151, 155, 157, 161, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 227, 233, 241, 249, 255

Offset: 1

Gus Wiseman, Dec 27 2018

nonn

A multiset multisystem is a finite multiset of finite multisets. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is uniform if all parts have the same size, regular if all vertices appear the same number of times, and strict if there are no repeated parts. For example, {{1,2,2},{1,3,3}} is uniform, regular, and strict, so its MM-number 13969 belongs to the sequence. Note that the parts of parts such as {1,2,2} do not have to be distinct, only the multiset of parts.

			The sequence of all strict uniform regular multiset multisystems, together with their MM-numbers, begins:
   1: {}           59: {{7}}         157: {{12}}        269: {{2,8}}
   2: {{}}         67: {{8}}         161: {{1,1},{2,2}} 271: {{1,10}}
   3: {{1}}        73: {{2,4}}       163: {{1,8}}       277: {{17}}
   5: {{2}}        79: {{1,5}}       165: {{1},{2},{3}} 283: {{18}}
   7: {{1,1}}      83: {{9}}         167: {{2,6}}       293: {{1,11}}
  11: {{3}}        85: {{2},{4}}     177: {{1},{7}}     295: {{2},{7}}
  13: {{1,2}}      93: {{1},{5}}     179: {{13}}        311: {{1,1,1,1,1,1}}
  15: {{1},{2}}    97: {{3,3}}       181: {{1,2,4}}     313: {{3,6}}
  17: {{4}}       101: {{1,6}}       187: {{3},{4}}     317: {{1,2,5}}
  19: {{1,1,1}}   103: {{2,2,2}}     191: {{14}}        327: {{1},{10}}
  23: {{2,2}}     109: {{10}}        199: {{1,9}}       331: {{19}}
  29: {{1,3}}     113: {{1,2,3}}     201: {{1},{8}}     335: {{2},{8}}
  31: {{5}}       123: {{1},{6}}     205: {{2},{6}}     341: {{3},{5}}
  33: {{1},{3}}   127: {{11}}        211: {{15}}        347: {{2,9}}
  41: {{6}}       131: {{1,1,1,1,1}} 227: {{4,4}}       349: {{1,3,4}}
  43: {{1,4}}     137: {{2,5}}       233: {{2,7}}       353: {{20}}
  47: {{2,3}}     139: {{1,7}}       241: {{16}}        367: {{21}}
  51: {{1},{4}}   149: {{3,4}}       249: {{1},{9}}     373: {{1,12}}
  53: {{1,1,1,1}} 151: {{1,1,2,2}}   255: {{1},{2},{4}} 381: {{1},{11}}
  55: {{2},{3}}   155: {{2},{5}}     257: {{3,5}}       389: {{4,5}}

Cf. A005117, A007016, A112798, A302242, A306017, A319056, A319189, A320324, A321698, A321699, A322554, A322703.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And[SquareFreeQ[#],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A323764 Dirichlet self-convolution of the integer partition numbers A000041.

Original entry on oeis.org

1, 1, 4, 6, 14, 14, 34, 30, 64, 69, 112, 112, 228, 202, 330, 394, 575, 594, 956, 980, 1492, 1674, 2228, 2510, 3700, 3965, 5276, 6200, 8126, 9130, 12318, 13684, 17842, 20622, 25808, 29976, 38377, 43274, 53990, 62976, 77912, 89166, 110656, 126522, 154918, 179744

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

Also the number of multiset partitions of constant multiset partitions of integer partitions of n.

			The a(4) = 14 multiset partitions of constant multiset partitions:
  ((1111))              ((22))      ((4))  ((31))  ((211))
  ((11)(11))            ((2)(2))
  ((11))((11))          ((2))((2))
  ((1)(1)(1)(1))
  ((1))((1)(1)(1))
  ((1)(1))((1)(1))
  ((1))((1))((1)(1))
  ((1))((1))((1))((1))

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A000837, A001970, A002033, A003238, A034729, A047968, A279787, A305551, A306017, A319056, A323774.

Join[{1},Table[Sum[PartitionsP[d]*PartitionsP[n/d],{d,Divisors[n]}],{n,1,100}]]

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019

A323776 a(n) = Sum_{k = 1...n} binomial(k + 2^(n - k) - 1, k - 1).

Original entry on oeis.org

1, 3, 7, 16, 40, 119, 450, 2253, 15207, 139190, 1731703, 29335875, 677864041, 21400069232, 924419728471, 54716596051100, 4443400439075834, 495676372493566749, 76041424515817042402, 16060385520094706930608, 4674665948889147697184915

Offset: 1

Gus Wiseman, Jan 27 2019

nonn

Number of multiset partitions of integer partitions of 2^(n - 1) whose parts are constant and have equal sums.

			The a(1) = 1 through a(4) = 16 partitions of partitions:
  (1)  (2)     (4)           (8)
       (11)    (22)          (44)
       (1)(1)  (1111)        (2222)
               (2)(2)        (4)(4)
               (2)(11)       (4)(22)
               (11)(11)      (22)(22)
               (1)(1)(1)(1)  (4)(1111)
                             (11111111)
                             (22)(1111)
                             (1111)(1111)
                             (2)(2)(2)(2)
                             (2)(2)(2)(11)
                             (2)(2)(11)(11)
                             (2)(11)(11)(11)
                             (11)(11)(11)(11)
                             (1)(1)(1)(1)(1)(1)(1)(1)

Seiichi Manyama, Table of n, a(n) for n = 1..120

Cf. A000123, A001970, A002577, A006171, A007716, A034729, A047968, A279787, A279789, A305551, A306017, A319056, A323766, A323774, A323775.

Table[Sum[Binomial[k+2^(n-k)-1,k-1],{k,n}],{n,20}]

a(n) = sum(k=1, n, binomial(k+2^(n-k)-1, k-1)); \\ Michel Marcus, Jan 28 2019

cp's OEIS Frontend

A323304 Heinz numbers of integer partitions that cannot be arranged into a matrix with equal row-sums and equal column-sums.

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A323348 Number of integer partitions of n whose parts cannot be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.

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A321699 MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.

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A321725 Irregular triangle read by rows where T(n,k) is the number of d X d non-normal semi-magic squares with d = A027750(n,k) and sum of all entries equal to n.

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A322531 Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

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A322703 Squarefree MM-numbers of strict uniform regular multiset systems spanning an initial interval of positive integers.

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A322789 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

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A322833 Squarefree MM-numbers of strict uniform regular multiset multisystems. Squarefree numbers whose prime indices all have the same number of prime factors counted with multiplicity, and such that the product of the same prime indices is a power of a squarefree number.

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A323764 Dirichlet self-convolution of the integer partition numbers A000041.

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